On linear integro-differential equations of Barbashin type in spaces of continuous and measurable functions
type: Item
instance of: scholarly article
title: On linear integro-differential equations of Barbashin type in spaces of continuous and measurable functions
zbMATH Open document ID: 0651.47040
DOI: 10.1216/JIE-1988-1-2-227
author: Jürgen Appell
author: O. W. Diallo
author: Peter P. Zabreĭko
published in: Journal of Integral Equations and Applications
publication date: 1988-01-01T00:00:00Z
review text: This paper surveys several important properties of linear integro- differential equations \[ (1)\quad du/dt=A(t)u \] of Barbashin type \[ (2)\quad A(t)x(s)=c(t,s)x(s)+\int^{b}_{a}k(t,s,\sigma)x(\sigma)d\sigma, \] especially those related to the geometric structure of the underlying function space. In contrast to Barbashin's classical results, also discontinuous data (e.g. kernel functions) are allowed. After discussing several classes of suitable kernels, the resolvent operator (Cauchy function) generated by the operator (2) is described. Moreover, stability results for equation (1) are proved. Finally, representation formulas for the corresponding Green's function are given; a perturbed version of such formulas applies to averaging procedures of Bogolyubov-Krylov type.
reviewed by: Jürgen Appell
Mathematics Subject Classification ID: 47Gxx
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 46E30
zbMATH DE Number: 4061998
zbMATH Keywords: linear integro-differential equatons
zbMATH Keywords: kernel functions
zbMATH Keywords: Cauchy function
zbMATH Keywords: stability results
zbMATH Keywords: representation formulas
zbMATH Keywords: Green's function
zbMATH Keywords: averaging procedures of Bogolyubov-Krylov type
MaRDI profile type: Publication
full work available at URL: jie-1988-1-2-227
OpenAlex ID: W1998581754
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description: scientific article; zbMATH DE number 4061998